In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. 7|5x| + 2 = 16

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations that involve it, as it can lead to two possible cases based on the definition.

Linear Equations

Linear equations are mathematical statements that express the equality of two linear expressions. They can typically be written in the form ax + b = c, where a, b, and c are constants. Solving these equations involves isolating the variable, which may require operations such as addition, subtraction, multiplication, or division. In the context of absolute value equations, linear equations arise from the cases derived from the absolute value definition.

Case Analysis

Case analysis is a method used to solve equations that involve absolute values by considering different scenarios. For an equation like |x| = a, where a is non-negative, two cases arise: x = a and x = -a. This approach is essential for finding all possible solutions to absolute value equations, as it ensures that both potential outcomes are examined, leading to a complete solution set.